1. Field of the Invention
The invention relates generally to flow rate measurement methods and systems. In particular, embodiments relate to multi-path ultrasonic measurement of partially developed flow profiles.
2. Background Art
In industries where a flow of fluid is involved, accurate measurements of flow rates are often required. For example, in the oil and gas industry, accurate flow rate measurements are needed in custody transfer (the transfer of ownership, for example at a crude oil loading and off-loading station), leak detection, and process control applications. Conventional flow measurement technologies include turbine flow meters and positive displacement flow meters. Recently, multi-channel ultrasonic meters are gaining shares in this market due to their advantages over conventional technologies. These advantages include: excellent long-term repeatability, less sensitivity to fluid properties such as viscosity and pressure, better open-box accuracy, wider range of linearity, and lower cost of maintenance due to the fact that no moving parts are used in these ultrasonic meters.
In a typical operation, an ultrasonic flow meter uses a transducer to transmit an ultrasonic beam into the flow stream, and the ultrasonic energy is received by a second transducer. The flow carrying the ultrasonic wave alters the wave's frequency (Doppler effect) and transit time (velocity superposition), and these two quantities can be measured to determine the flow rate. Based on these principles, two major ultrasonic flow measurement technologies exist: Doppler and transit-time. In some configurations of flow meters, transducers are clamped on the outside wall of a pipe. To achieve better measurement accuracy, transducers may be alternately placed inside the pipe wall, and such transducers are referred to as “wetted” transducers. Some methods to measure flow profiles have been developed, mostly based on Doppler technology (e.g., U.S. Pat. Nos. 6,067,861, 6,378,357). However, Doppler signals rely heavily on particle size and concentration that sometimes may vary and lead to poor repeatability. It is widely accepted in the industry that only multi-path transit-time meters combined with wetted transducers are capable of high accuracy applications mentioned above.
The principles of transit-time ultrasonic measurements are well established. According to the an America Petroleum Institute (API) standard (API H00008, Manual of Petroleum Measurement Standards, Measurement of Liquid Hrdrocarbons by Ultrasonic Flowmeters Using Transit Time), an average velocity along an ultrasonic path can be derived from:
                              V          i                =                              L                          2              ⁢                                                          ⁢              cos              ⁢                                                          ⁢              θ                                *                                                    t                2                            -                              t                1                                                                    t                1                            *                              t                2                                                                        (        1        )            where Vi is a path-average flow velocity (i.e., an average of velocities along a particular ultrasonic path) for the path i, L is the ultrasonic path length, θ is the angle between the ultrasonic path and the fluid velocity vector, and t1 and t2 are the ultrasonic travel times in and against the flow direction, respectively.
It should be noted that the measured path-average velocity Vi is different from the flow-average velocity Vavg, the latter being the velocity averaged over the flow cross section. Vi is directly measured from an ultrasonic transit-time flow meter using Equation (1), while Vavg gives the flow rate that is important in applications such as custody transfer. A ratio Ki between these two velocities can be defined as:
                                          K            i                    =                                    V              avg                                      V              i                                      ,                            (        2        )            and is referred to as the channel factor. In the following description, Vi is referred to as the path velocity, and Vavg is referred to as the average velocity.
Pipe flows mostly run in one of the two modes: the laminar mode and the turbulent mode. Widely accepted mathematical models for these flow modes are:
                                          V            ⁡                          (              r              )                                =                                    V              c                        *                          (                              1                -                                                      r                    2                                                        R                    2                                                              )                                      ,                  for          ⁢                                          ⁢          laminar          ⁢                                          ⁢          flow                                    (        3        )                                                      V            ⁡                          (              r              )                                =                                    V              c                        *                                          (                                  1                  -                                      r                    R                                                  )                                            1                N                                                    ,                            (        4        )                                for        ⁢                                  ⁢        turbulent        ⁢                                  ⁢        flow        ⁢                                  ⁢        in        ⁢                                  ⁢        smooth        ⁢                  -                ⁢        wall        ⁢                                  ⁢        pipes                                        where V(r) is the velocity at a distance r from the pipe centerline, R is the pipe radius, Vc is the flow velocity along the pipe centerline, and N is a power-law factor.
The power-law factor N is a characteristic value of a turbulent flow. For a fully developed turbulent flow, the power-law factor N can be estimated using an empirical equation described in the literature (e.g., L. Lynnworth, “Ultrasonic Measurement for Process Control”, Academic Press, San Diego, 1989):N=1.66*log Re  (5)where Re is the Reynolds number, which is a function of the flow velocity V and the fluid viscosity μ:
                              Re          =                                    DV              ⁢                                                          ⁢              ρ                        μ                          ,                            (        6        )            where D is the pipe diameter, and p is the fluid density.
In real-world applications, the pipe and the fluid conditions often cannot be precisely quantified, and Equations (5) and (6) generally cannot be used to obtain N in high-accuracy measurements. Hence, for a given turbulent flow, at least two measurements at two flow paths are needed in order to solve Equation (4) for the two unknowns, N and Vc. This is why a multi-path ultrasonic technology is often needed to resolve flow profile variations.
Referring to FIG. 1A, a pipe 1 is shown with three pairs of transducers, 11a and 11b, 12a and 12b, 13a and 13b. An arrow 2 shows the flow direction. The lines between transducer pairs show their ultrasonic paths. In this configuration, the ultrasonic path between 11a and 11b crosses the pipe centerline, and is referred to as a diagonal path. The shortest distance from a path to the pipe centerline is referred to as channel level. A diagonal path has a channel level of 0. The path between 12a and 12b and the path between 13a and 13b have the same channel levels h even though the two transducer pairs are at different locations. FIG. 1A also illustrates an exemplary flow profile 3.
The channel factor Ki depends on flow profiles and the position of the ultrasonic path. For a diagonal ultrasonic path the channel factor is:
                              0.75          ,                                    for              ⁢                                                          ⁢              a              ⁢                                                          ⁢              laminar              ⁢                                                          ⁢              flow                        ;            and                          ⁢                                  ⁢                                                            ∫                0                R                            ⁢                                                                    (                                          1                      -                                              r                        R                                                              )                                                        1                    N                                                  *                2                *                r                *                                                                  ⁢                                  ⅆ                  r                                                                    R              *                                                ∫                  0                  R                                ⁢                                                                            (                                              1                        -                                                  r                          R                                                                    )                                                              1                      N                                                        *                                                                          ⁢                                      ⅆ                    r                                                                                ,                      for            ⁢                                                  ⁢            a            ⁢                                                  ⁢            turbulent            ⁢                                                  ⁢                          flow              .                                                          (        7        )            
Referring to FIG. 1B, the channel factor K (vertical axis) is shown as a function of Reynolds number Re (horizontal axis) in a case of a diagonal ultrasonic path. For non-diagonal ultrasonic paths, the K values can also be derived in a similar way. The relation between K value and path positions has been well studied for the laminar flow and the turbulent flow profiles, dating back to a 1978 U.S. Pat. No. 4,078,428. Hence, for both laminar flow and turbulent flow profiles, the relations between measured velocities and the actual average velocities are well defined.
A major challenge for ultrasonic meters is to detect a flow profile promptly based on information from a limited number of paths. For a turbulent flow, the randomness of measured path velocities in selected paths can result in instant deviation as high as 10 percent from the average values, and the small number of paths makes it difficult to obtain a satisfactory statistical average flow velocity. Unlike turbine meters that inherently average out the whole cross section of a flow, transit-time ultrasonic flow meters only measure a limited number of selected paths of flow velocity. To average out randomness of the measured velocities, ultrasonic meters need to have either a large damping on the raw data or more paths distributed across a flow profile. Using a large damping will adversely affect the system response time and result in poor repeatability when measuring a small volume. On the other hand, adding more channels to measure more paths may substantially increase the system cost.
A more challenging problem for ultrasonic flow meters is to detect a partially developed flow profile. A fully developed flow profile, by definition, is a flow velocity distribution pattern that does not change along a pipe. Any other flow profiles that have symmetric velocity distributions around the pipe centerline, but with a evolving flow velocity distribution along the pipe, are in this description referred to as partially developed flow profiles. There are two possible causes for partially developed flow profiles. One is a transitional profile between turbulent and laminar flows that can happen in high-viscosity fluids. Another is due to the presence of a flow conditioning device that does not have enough downstream length for the profile to fully develop.
The transition between a turbulent flow and a laminar flow normally takes place when the Reynolds number is around 2300 and has been demonstrated by numerous experiments. However, as illustrated in FIG. 1B, this transition can happen in a wide range of Reynolds numbers and can have memory effects, depending on fluid and pipe conditions. As a result, it is not accurate to use the Reynolds number as a sole indicator of a flow profile mode. A partially developed profile near the transition range may cause an unacceptably poor repeatability in flow measurements, as neither the laminar nor the turbulent model fits the profile well.
As a known fact, a steady profile needs some straight, obstruction-free distance in the pipe to fully develop. Referring to FIG. 2, a plug flow 21, which has a constant velocity throughout the cross section of the pipe 23 having a diameter D, enters the pipe 23 from a much larger pipe 22. The flow initially has a partially developed profile 24. After an entrance length 25, the flow has a fully developed profile 26. Both theories and experiments indicate that the entrance length 25 needs to be as long as 100 times the pipe diameter D for a laminar profile to fully develop, and 80 times the pipe diameter D for a turbulent profile to fully develop (see, R. W. Fox and A. T. McDonald, “Introduction to Fluid Mechanics”, 3rd ed., John Wiley and Sons, New York, 1992). In practice, entrance flow rarely has a form of a plug profile, and a straight length of 10 to 15 times the pipe diameter D is commonly recommended by meter manufactures to have a predictable measurement of flow profiles.
For a multi-path custody-transfer flow meter, it is a common practice to have a flow conditioning device installed upstream of the flow meter. The main purpose of the flow conditioning device is to reduce swirls and to reduce asymmetric profile distortion. A shorter flow conditioning installation distance is always beneficial for manufacturers and customers. U.S. Pat. No. 6,647,806 also suggests that a shorter distance between a flow conditioning device and a flow meter can improve the repeatability of measurements.
Referring to FIG. 3, which shows an API recommended flow conditioning device 31, made of a bundle of small tubes having a length B, is installed in the pipe 32 at a distance A from the pipe entrance. The distance C downstream of the flow conditioning device 31, is recommended to be at least 5 times the diameter D of the pipe 32. The device 31 tends to smooth out a flow profile. As a result, the presence of the device 31 will either disturb a laminar profile, or flatten a turbulent profile. In either case, within a limited entrance distance, a typical flow profile (i.e., laminar flow or turbulent flow) may not fully develop. In particular, downstream of the conditioning device, turbulent flow may have a much higher N number than that estimated using equation (5), and turbulent-laminar transition can happen at a Reynolds number much lower than 2300. In either case, the flow profile will be unpredictable based on the Reynolds number. What are still needed are improved systems and methods for monitoring partially developed flow profiles in real time.